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The Time Dependent Traveling Salesman Planning Problem inControlled Airspace
Fabio FuriniLAMSADE, Universite Paris-Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris, France
Carlo Alfredo PersianiENAV S.p.A, Italian Agency for Air Navigation Services Via Salaria 716 Roma, Italy
Paolo TothDEI, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy
Abstract
The integration of Unmanned Aircraft Systems (UAS) into civil airspace is one of the most challenging
problems for the automation of the Controlled Airspace, and the optimization of the UAS route is a key
step for this process. In this paper, we optimize the planning phase of a UAS mission that consists of
departing from an airport, flying over a set of mission way points and coming back to the initial airport.
We assume that during the mission a set of piloted aircraft flies in the same airspace and thus the cost of the
UAS route depends on the air traffic and on the avoidance manoeuvre used to prevent possible conflicts.
Two Air Traffic Management techniques, i.e., rerouting and holding, are modelled in order to maintain
a minimum separation between the UAS and the piloted aircraft. Heuristic algorithms are proposed for
the solution of the considered problem, called the Time Dependent Traveling Salesman Planning Problem
in Controlled Airspace (TDTSPPCA). A mathematical formulation based on a particular version of the
Time Dependent Traveling Salesman Problem (TDTSP), which allows holdings at mission way points, is
proposed for solving the TDTSPPCA, and applied to the UAS route planning phase to minimize the total
operational cost. Another formulation, based on a Travelling Salesman Problem variant that uses specific
penalties to model the holding times, is proposed and compared with the first one. Finally, computa-
tional experiments on real-world air traffic data from Milano Linate Terminal Manoeuvring are reported
to evaluate the performance of the proposed models and of the heuristic algorithms.
Keywords: Integer Programming, Traveling Salesman Problem, Time Dependence, Unmanned Aerial
Systems, Air Traffic Management
Preprint submitted to Transportation Research Part B June 21, 2015
1. Introduction
The Traveling Salesman Problem (TSP) is one of the most investigated problems in Transportation
Science and many applications and variants have been studied during the last few decades. The temporal
extension of the TSP, called Time Dependent Traveling Salesman Problem (TDTSP), has been addressed
to deal with time dependent problems as traffic operations management in congested urban areas. The
basic idea is that the cost of a path between two locations depends on the time of the day: during peak
hours, the traversing time could significantly change with respect to different hours of the day. The ver-
sion of TDTSP arising in Controlled Airspace presents three main differences with respect to the version
of TDTSP concerning applications in urban areas. First, the sampling interval must be compatible with
the characteristics of the air traffic, i.e., the temporal discretization must consider minutes instead of hours.
Secondly, the international regulation of the airspace defines a minimum separation that must be applied
between aircraft to avoid conflicts. The third difference concerns the fact that, in Controlled Airspace,
paths between locations are forbidden during specific time periods due to the presence of other aircraft.
These important differences lead to different mathematical models and solutions techniques.
There are several applications of the TDTSP in Controlled Airspace: one of the most important is the route
planning for automated aircraft. The use of Unmanned Aircraft Systems (UAS) in military missions has
proved their capability and versatility in different applications. The so called “D3 paradigm” (Dull, Dirty,
Dangerous) emphasizes the ability of UAS to operate autonomously in complex scenarios, reducing both
the necessity of human intervention and the correlated risks. There are many civil applications that can
also be performed by UAS: for example, crop spraying or active volcano monitoring as “dirty” missions,
crowd and traffic monitoring as “dull” missions, and search and rescue (SAR), fire monitoring and fire
fighting as “dangerous” missions. The D3 paradigm can be translated in civil contexts, achieving several
advantages like reducing risk for human crew, increasing mission efficiency and, not less important, re-
ducing operational and personnel costs. As a case study in our computational tests, we use the planning
phase of a so called “Intelligence, Surveillance, and Reconnaissance” (ISR) mission for UAS. We choose
this application since the UAS operations are at a higher level of automation than those associated with
the piloted aircraft. However, the proposed methodology could be applied also to the piloted traffic when
the level of automation in the traffic control increases.
The Time Dependent Traveling Salesman Planning Problem in Controlled Airspace (TDTSPPCA) con-
sidered in this paper consists of finding the best flying sequence of an aircraft over a set of fixed mission
way points at a given altitude, aiming at minimizing the operational cost and, at the same time, ensuring
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a minimum separation with respect to the planned traffic. Two avoidance techniques are considered: the
possibility of rerouting with respect to the optimal TSP sequence, and the possibility of holding the aircraft
over a mission way point. These avoidance techniques are particularly effective in the planning phase as
the one considered in this work.
Paper contributions.. We propose two new Mixed Integer Linear Programming (MILP) formulations of
the Time Dependent Traveling Salesman Planning Problem in Controlled Airspace: the Time-Based model
and the Penalty-Based model. Both formulations are capable of successfully dealing with real-world data,
and specific algorithms are developed to speed up the solution time. Our methodology is based on a hybrid
approach that merges Air Traffic Management (ATM) and Operations Research techniques. This approach
shows that the problem of finding the best flying sequence to visit a set of mission way points in terms
of total cost and the problem of maintaining a minimum separation between the aircraft can be combined
using a temporal extension of the classical Traveling Salesman Problem (TSP). In other words, our goal
is to solve the UAS ATM planning phase by proposing an approach able to find optimized conflict-free
routes. The output of the planning phase can then be the input for further tactical phases as the trajectory
optimization or the real-time reoptimization of the routes (see Section 8 for further details on this issue).
The paper is organized as follows. In Section 1.1 we describe the mission scenario, its constraints and
the assumptions used. In Section 2 we describe the literature on related topics. In Section 3 we model
the mission environment and the separation constraints. In Section 3.1 we formally address the Traveling
Salesman Planning Problem in Controlled Airspace and, in Section 4 we present new heuristic algo-
rithms. In Section 5, we introduce the Time-Based model and, in Section 6, the Penalty-Based model.
Finally in Section 7, we report extensive computational experiments testing the proposed formulations
and the heuristic algorithms on real air traffic data from Milano Linate (LIML) Terminal Maneuvering
Area (TMA). In Section 8, we discuss possible extensions of the models in order to take into consideration
additional real-world features of the problem and the conclusions are discussed in Section 9.
1.1. Aeronautical background
The Controlled Airspace is the part of the airspace where the air traffic control service is provided, i.e.,
the air traffic controllers ensure a minimum spatial and/or temporal separation between aircraft. For this
aim, they control for each aircraft in their area of responsibility:
• route,
• altitude,
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• speed,
• rate of climb/descent,
• heading.
A pilot plans his flight from an airport to another one requesting a route and a cruise level to the air traffic
control. The air traffic control approves the request or re-plans it. The route of an aircraft is defined by a
sequence of navigation points and altitudes. An airway is a predefined succession of navigation points. An
aircraft can follow an airway or can proceed directly between two navigation points belonging to different
airways if cleared by the air traffic control. The goal of the air traffic control is to ensure a safe and ordered
flow of the air traffic in the Controlled Airspace. The controllers provide the legal minimum separation
between aircraft using different techniques, the most used ones are:
• the vectoring,
• the speed control,
• the use of holdings,
• the rerouting,
• the use of different levels.
The use of a particular technique depends on several factors as:
• the airspace layout,
• the aircraft performances,
• the human factors,
• the pilot training.
The choices of the separation technique, of the routes and of the altitudes are strictly connected and affect
the controlled air traffic and its evolution.
This is the ideal general context of a routing problem in Controlled Airspace. Moreover, there are many
stochastic aspects that air traffic controllers have to face during the traffic management, e.g. the real traffic
may differ from the planned one and the weather conditions affect the routes and the avoidances. Since
considering these aspects all together goes behind the scope of the paper, we put ourself in the context of
planning a conflict free route considering the following realistic assumptions.
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1.2. Modeling assumptions
We investigate what is known in the literature as the static case of the problem.
• We assume that the air traffic the UAS may encounter is known in advance. We take into account the
real air traffic of a simulation day in a specific Terminal Area (i.e., a sector of Controlled Airspace
around big airports) using historical data regarding the aircraft routes. We simulate such a traffic
plan for the UAS mission, i.e., we optimize the UAS planned route, as it had flown in such an
environment.
• We consider the UAS as traffic with lower priority; this is a realistic assumption since some UAS
missions in Controlled Airspace will require a flexible management of the automated aircraft.
We assume that the conflict detection is performed in the planning phase using foreseen air traffic
data and not from a real-time perspective. Further developments, especially regarding the dynamic
case, will require the assumption that the UAS can detect other aircraft.
• The real air traffic is composed by the VFR (Visual Flight Rules) traffic, i.e., the aircraft that navigate
by sight, and by the IFR (Instrumental Flight Rules) traffic, i.e., the aircraft that navigate using
instruments. We consider only the IFR traffic.
• We model only some avoidance techniques, i.e., the rerouting and the use of holdings, obtaining a
conflict free optimized route from a planning perspective. Such a route can be post-optimized in
order to define the trajectory in a given scenario.
• Finally, we assume that once an holding has been planned, the UAS can complete it without addi-
tional separation conflicts (e.g., in case of conflict the altitude can be changed accordingly).
2. Related works
Given a list of points and the distances between each pair of them, the TSP consists of finding the
shortest possible Hamiltonian circuit, i.e. the minimum distance route that visits each point exactly once.
The TSP has been widely studied in the past years and many exact and heuristic algorithms have been
developed to solve different routing problems. For a complete survey on the TSP see [9]. However,
studies on TSP with time-dependent travel costs (TDTSP) are fairly rare in the literature. Two main kinds
of TDTSP have been proposed in the literature. The first one is characterized by arc costs that depend on
the position of each arc in the Hamiltonian circuit, the second one by arc costs that depend on the time at
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which each arc is visited. As far as the first kind of TDTSP is concerned, we refer the interested reader to
the seminal work of [17].
The second kind of TDTSP arose to deal with traffic jams in congested urban areas: during peak
periods the cost of a path between two clients becomes higher than in normal conditions. We follow the
second stream of research as this model is also useful to interpret air traffic using a convenient time horizon
compatible with aircraft operations. In fact, the cost of a UAS route between two mission way points
depends on the air traffic met by the UAS, starting a path at a specific time. Considering a time interval
of one minute and a mission duration of some hours many exact algorithms proposed in the literature
are not able to provide solutions in an acceptable computing time. [3] proposed a heuristic algorithm
for the TDTSP, where two periods of the day present different travel costs, adapting the classical savings
algorithm developed by Clark and Write for the Vehicle Routing Problem. [11] proposed a Mixed Integer
Linear Programming (MILP) formulation of the TDTSP and of the related time dependent vehicle routing
problem TDVRP; they handled the travel cost function as a step function and presented several heuristic
algorithms to solve these problems.
In [8] a classification of the TDTSP is presented, considering old and new formulations obtained from a
quadratic assignment model for the TDTSP. [12] used a dynamic programming heuristic algorithm to solve
the TDTSP with a given starting time from the depot. [15] considered the bi-criteria TDVRP with time and
area dependent travel speed. The minimization of the total vehicle operation time and the minimization
of the total weighted tardiness were the conflicting objectives studied, and a MILP formulation for the
problem was presented. This model was characterized by the waiting time at nodes that occurs when a
vehicle awaits the next time interval for more rapid movement. Park also presented a heuristic algorithm
called the bi-criteria-saving algorithm to solve this problem. [7] described the derivation of travel time
data from traffic information systems. They also tried to overcome the problem of the so-called non-
passing property using a smoothed travel time function for calculating the arrival time given a departure
time. The non-passing property arises in the TDVRP when an earlier departure time of a vehicle must
be coupled with an earlier arrival time, and vice versa. Computational results were reported on instances
based on traffic information obtained from the city of Berlin. [4] solved the real-time TDVRP with time
windows, using a heuristic algorithm that included methods for route construction; a technique to choose
the optimal departure time was also developed. [2] considered a version of TDTSP with time windows
proposing an exact solution through a graph transformation. Such a problem is first transformed into
an Asymmetric Generalized TSP and then into a Graphical Asymmetric TSP in order to apply exact
known algorithms for the Mixed General Routing Problem. [19] proposed a branch and cut algorithm
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designed for a TDTSP used to model a production scheduling problem. In fact, since the set-up time
between two jobs is a function of the completion time of the first job, the considered scheduling problem
can be reformulated using a TDTSP approach. [19] also introduced some families of valid inequalities
used to strengthen the Linear Programming Relaxation of the proposed Integer Linear Programming (ILP)
formulation. [18] considered the TDVRP with time windows; they showed how to transform this problem
into an Asymmetric Capacitated Vehicle Routing Problem in order to solve it with known exact or heuristic
algorithms. [10] proposed a Tabu Search algorithm to solve the TDVRP and the related goods assignment
problem: a real-world case of a warehousing company was used to illustrate the studied method. In [1], a
study of the polytope associated with a particular TDTSP formulation is presented and a branch-cut-and-
price approach is applied to instances from the TSPLIB. Finally, a routing problem for UAS in controlled
airspace has been presented in [16], where the authors propose a Genetic Algorithm designed to manage a
UAS mission in civil airspace considering all the avoidance techniques previously mentioned.
3. Airspace and traffic separation modeling
The mission environment is modeled considering its key features: the air space layout, the aircraft and
their routes, the UAS and the ATC avoidance techniques. The time is discretized, i.e. we consider the time
as a succession of time steps separated by an interval chosen accordingly to the precision necessities. If
tmax is an upper bound on the mission duration obtained by the UAS endurance and ∆t is the duration
of a time step, the set T = {0, 1, 2, . . . , nts} is the set of time steps where nts = tmax/∆t represents the
number of time steps used.
We consider a set A of n aircraft flying during the period under investigation and an airspace defined by
the set NA of q navigation points used by the aircraft. Each navigation point p ∈ NA is characterized by
a geographical position. Each aircraft a ∈ A is characterized by a route ra ⊆ NA defined by a succession
of navigation points. The altitude and the time of the aircraft a passing through the navigation point p
are called respectively lap and tap, moreover we define the set VA as the set of all the pairs of navigation
points and the set RA = ∪a∈Ara as the set of all aircraft routes. Finally we introduce the complete graph
GA = {NA, VA} as the ”Airspace Graph”, i.e. the graph that defines the aircraft routes. This graph
is directed and represents the airspace environment. Each aircraft, which flies in the modeled airspace
during the mission, is moving over the Aircraft Graph GA.
The surveillance mission consists of departing from an airport (mission way point 0), taking pictures
over a set M \ 0 of (m − 1) mission way points and coming back to the airport. Each mission way point
i ∈ M is characterized by a geographical position and an altitude. We assume that between each pair of
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mission way points a direct route is available; let VU be the set of all pairs of mission way points and we
denote a specific pair with (i, j). We introduce the complete graph GU = {M,VU} as the ”UAS Graph”
,i.e. the graph that defines the UAS routes.
A holding pattern is available over each mission way point: in other words the UAS can overfly and,
if necessary, can perform one or more orbits over a mission way point. We assume that the cost of a route
can be approximated using the flight time, once the distances between the mission way points and the
UAS mission parameters are known. These parameters (called Puas) depend on the UAS performance and
consist of the speed vU (best range speed or best endurance speed, cruise speed), the rate of climb rc and
the rate of descent rd that the UAS would have used in the same routes in segregated air space. Using
such parameters, we define the flight time fij necessary to reach the mission way point j from the mission
way point i. Due to the different performance during the climb and descent phases, we assume that the
time from i to j is different from the time from j to i. Finally, since we assumed the UAS as the airspace
user with lower priority, if a conflict between an aircraft and the UAS exists, the UAS has to wait or to
change its route and not vice versa (see [13] for further details on this topic). According to the air space
classification, a minimum separation must exist between each pair of aircraft (UAS included); two ATC
separation techniques are modeled and applied to the UAS planned route: the rerouting or the holding over
a mission way point.
Let S be the set of all possible conflicts: a conflict s ∈ S between the UAS sub-route (i, j) ∈ VU , and
an aircraft route ra occurs if at least one point exists on the UAS sub-route such that its distance from ra
is less than 1000 ft on the vertical plane or 5 NM on the horizontal plane. In this case, for the conflict s
we define the Conflict Point psc as the point closest to ra. The time step in which the aircraft is overflying
the closest point to psc on its route is defined as the Conflict Time Step tsc. The avoidance of a conflict is
modeled by forbidding the departure of the UAS from the mission way point i to the mission way point j
in the time steps involved in such conflict. For a given pair of UAS sub-route (i, j), and aircraft route ra,
a time step is involved in a conflict s if a UAS departure in such time step would determine an insufficient
separation. Let us indicate withXsij (Xs
ij ⊆ T ) the conflict interval as the interval defined by the time steps
involved in the conflict s over the sub-route (i, j).
The identification of the time steps involved in a conflict for a UAS sub-route can be performed using
simple geometric considerations: we use the conflict detection geometric algorithm presented in [16]
where it is also reported a complete analysis of all possible conflict cases. We report in Figure 1 the general
case of conflict. The Conflict Zone consists of a sub-separation area defined by the UAS and the aircraft
route segments that are not separated from each other. Each conflict has an associated Conflict Zone: the
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Conflict Zone can not be occupied by the UAS and an aircraft at the same time. Once checked that the
Figure 1: Conflict Zone
vertical separation of 1000ft does not exist between all points of the conflicting routes, their projection
on the horizontal plain is used to identify the Conflict Zone. The segments (pi, pj) and (pa, pb) represent
the projection on the plane of the UAS sub-route (i, j) and of the aircraft route (a, b), respectively. ta and
tb are the time steps in which the aircraft is overflying pa and pb respectively. The point p1cu is defined
as the projection of the first point over the UAS sub-route between pi and psc that presents a distance
from the aircraft route of 1000 ft in the vertical direction or 5 Nautical Miles (NM) in the horizontal
direction, whichever happens first, moving from pi to psc. Similarly, the point p2cu is the projection of the
first point over the UAS path between pj and pc that presents a distance from the aircraft route of 1000 ft
in the vertical direction or 5NM in the horizontal direction, whichever happens first, moving from psc to
pj . Analogously the points p1ca and p2ca are defined over the route of aircraft a; the time steps in which
aircraft a overflies such points are, respectively, t1ca and t2ca. The Conflict Zone is defined by the points
p1cu, p1ca, p2cu and p2ca and it is comprised by two ”sub-separation stretch”: the first one is identified over
the UAS route between p1cu and p2cu and the second one over the aircraft route between p1ca and p2ca. If
the UAS is moving from pi to pj and the aircraft from pa to pb, the Conflict Interval over pi starts after the
last departing time step available before the conflict. That is the departing time step from pi such that the
UAS can arrive in p2cu when the aircraft a is in the point p1ca. The holding starts after the time step t′pi:
t′pi = max{0, ta + ∆tpa,p1ca −∆tpi,p2cu} (1)
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Where ∆tpi,p2cu is the time step interval necessary for the UAS to fly from pi to p2cu maintaining Puas,
and ∆tpa,p1ca is the time step interval necessary to the aircraft to fly from pa to p1ca. In the same way, the
Conflict Interval over pi finishes before the first re-departing time step available after the conflict. That is
the departing time step from pi such that the UAS can arrive in p1cu when the aircraft a is in the point p2ca.
The holding finishes before the time step t′′pi:
t′′pi = max{0, ta + ∆tpa,p2ca −∆tpi,p1cu} (2)
Where ∆tpa,p2ca is the time step interval necessary for the aircraft to fly from pa to p2ca, and ∆tpi,p1cu is
the time step interval necessary for the UAS to fly from pi to p1cu. The dimension of the Conflict Zone
depends on the angle α (that defines if the routes are diverging or converging) and on the altitudes of the
UAS and the aircraft on the mission way points. If α = 90 and the altitudes on the points are the same,
the Conflict Zone consists of a circle. In case α = 180 (opposite routes) or α = 360 (same routes) and
the altitudes on the points are the same, the entire sub-route becomes a Conflict Zone during the aircraft
passage. Using the notation reported above it is possible to write the conflict interval Xsij defined by the
time steps involved in the conflict s over the UAS sub-route (i, j):
Xsij =]t′pi , t
′′pi
[ (3)
Considering all the conflicts over the UAS sub-route (i, j), we indicate with Xij the set of the all time
steps involved, i.e. the time steps in which a departure from mission way point i to the mission way point
j produces a conflict with an aircraft a ∈ A:
Xij = ∪s∈SXsij (4)
Considering that the air traffic and the related avoidance techniques used by the UAS in a sub-route (i, j)
depend on the time at which the UAS starts to fly that sub-route, the problem combines two decisions: the
choice of the best order of visiting a set of mission way points and the choice of the avoidance techniques
that have to be used. These two dimensions can be combined in a Time Dependent Asymmetric Travelling
Salesman Problem where the cost of a path between two mission way points varies in time. Starting a sub-
route at a specific time could produce a conflict and thus require appropriate management, i.e. rerouting
or holding.
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t
Wij(t)
t′pi ∆t t′′pi
fij + (t′′pi − t′pi
)
fij
Figure 2: The time step function
Figure 3 describes the traversing time (called Wi,j(t) in the figure) of the sub-route (i, j) as a function
of the departing time step t. Wi,j(t) can be calculated considering that at each time step t the UAS has
two options to proceed from i to j: if no conflict occurs (t /∈ Xi,j) the UAS can proceed directly from i to
j and thus the traversing time corresponds to the flight time fi,j , while in the event of conflict (t ∈ Xi,j)
it consists of the sum of two components: the remaining holding time from t until the first no conflicting
time step and the flight time fi,j .
3.1. Problem definition
Using the previous notation, we define the Time Dependent Traveling Salesman Planning Problem in
Controlled Airspace (TDTSPPCA) as follows: let G = {Gu, A,Ga, S, T} be the mission conflict system,
where at each mission way point i ∈M , a holding pattern is permitted. Let βi be the cost of a time step in
the holding pattern associated with i, and γ the cost of a time step during the flight phase. Moreover, for a
given set of flight parameters Puas, let fi,j be the flight time from point i to point j, and kmin the minimum
duration of a holding and kmax the maximum duration of a holding (with kmax ≥ kmin). For each pair
(i, j) of mission way points, let Xi,j be the set of time steps in which a departure from point i to point j
produces a conflict with an aircraft a ∈ A. The goal consists of finding the route starting and coming back
to the same prefixed mission way point (airport) that minimizes the mission cost, defined by the weighted
sum of the flying time and the holding time, maintaining the minimum separation from the piloted traffic
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and visiting all the mission way points.
The value of the optimal solution of the ATSP associated with the graph GU represents a lower bound
on the value of the optimal solution of the TDTSPPCA defined by the formulation (5)-(17) described
in Section 5.1, while the value of the solution defined by the heuristic algorithm described in Section 4,
called heuristic based on the ATSP route (HTSP), obtained by following the ATSP route and performing
the appropriate holdings represents instead an upper bound on the value of the optimal solution of the
TDTSPPCA. The TDTSPPCA is an asymmetric problem for two reasons: (i) the cost (flight time) of
the sub-route (i, j) may be different from the cost of the sub-route (j, i) because the related mission way
points may be at different altitudes; (ii) due to the presence of the air traffic, if the cost of the sub-routes
is the same in both directions (symmetric TSP) two tours that present the same sub-route but in opposite
directions could have different total times. Moreover, typical heuristic approaches for the TSP as the
k − opt exchange or insertion heuristics cannot easily be extended to the TDTSPPCA (see [11]). Another
important characteristic of the problem follows from the properties of the considered objective function. If
the holding costs βi, (i ∈M) are all equal since it is not important where the UAS waits, once a sequence
of mission way points is fixed, the best way to visit them consists of waiting as less as possible over each
of them. In other words, the route cost depends only on the sequence of the mission way points. This
obviously does not hold when the holding costs are different.
4. Heuristic Algorithms
The nature of the considered problem is such that the air traffic situation changes at least every minute;
this is a significant difference with respect to the TDTSP developed for taking into account the traffic in
big urban areas where the update of the traffic can be done every hour or couple of hours. This implies that
in a reference period of one day, the time steps in a city traffic problem are in a small number with respect
to those corresponding to a 4 hours endurance UAS mission with a time step of 1 minute. Moreover, since
tmax is an upper bound defined using the UAS endurance, it may be quite long with respect to the real
mission duration. An appropriate tailoring of the set T becomes critical to solve the TDTSPPCA and thus
heuristic approaches can be used to properly reduce the size of the set T .
Two heuristic algorithms have been designed to this aim: a Nearest Neighbour algorithm (NN) and an
ATSP route based algorithm (HTSP), plus a local search algorithm designed to further improve the solu-
tions obtained.
The ATSP route based algorithm (HTSP) starts from the optimal TSP on the graphGu and the resulting
tour is copied in the route vector. The holding vector is obtained by following the route defined by the
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route vector and performing the appropriate holdings: if a departing time step from a mission way point
belongs to a Conflict Interval the holding corresponds to the wait until the first available departing time
step (otherwise the holding is set to 0). In the Nearest Neighbour algorithm (NN), the solution vector is
initially void and the algorithm starts initializing the route vector with the airport. The solution is then
iteratively built, searching for the nearest mission way point in terms of flight time and holding cost (total
cost). Once a mission way point is inserted into the route vector, the related holding to reach it is inserted
in the holding vector.
The neighbourhood used for the Local Search algorithm (LS), is the set of solutions obtained by perform-
ing all the exchanges between each pair of mission way points. The Local Search algorithm is initialized
through the route provided by one of the two previously mentioned heuristic algorithms. If a cheaper route
can be found in the neighbourhood the current solution is updated and the algorithm reiterates, otherwise
the algorithm terminates with the current best solution. Since the described algorithms require a very short
computing time, a good upper bound tub on the mission duration may be fastly computed as the smallest
between their solutions. Using this upper bound the number of time steps nts is possibly reduced. Detailed
results of the heuristic algorithms are reported in Section (7).
5. Time-Based Model
In this section we present a time extension of a classical ATSP formulation, a Variable Fixing Procedure
to reduce the dimension of the model and a Branch and Cut Algorithm.
5.1. Mathematical Formulation and main features
Two types of binary variables are used to render the time dimension of the problem (variables xtij) and
to model the holding over the mission way points (variables yti). Variable xtij ((i, j) ∈ VU , t ∈ T ) has value
1 if the UAS starts to fly from mission way point i to mission way point j at time step t and 0 otherwise.
Variable yti (i ∈ M, t ∈ T ) takes value 1 if the UAS arrives at the mission way point i at time step t and 0
otherwise. By using these variables, for each mission way point i ∈M , the arrival and departure times are
given, respectively, by∑
t∈T yti and
∑t∈T
∑j∈M xtij . The Time-Based model of the TDTSPPCA reads as
follows:
min γ∑t∈T
∑(i,j)∈VU
fijxtij +
∑i∈M
βi(∑t∈T
∑j∈M
txtij −∑t∈T
tyti) (5)
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∑t∈T
∑j∈M
xtji = 1 i ∈M (6)∑t∈T
∑j∈M
xtij = 1 i ∈M (7)∑t∈T\{0}
yti = 1 i ∈M \ {0} (8)
∑j∈M,t−fji≥0
xt−fjiji ≤ yti i ∈M \ {0}, t ∈ T (9)
∑j∈M
xtij ≤t∑
t∗=0
yt∗
i i ∈M, t ∈ T (10)∑t∈T
∑j∈M
txtij −∑t∈T
tyti ≥ kmin ∀i ∈M (11)∑t∈T
∑j∈M
txtij −∑t∈T
tyti ≤ kmax ∀i ∈M (12)
xtij = 0 t : t+ fij ≥ tmax, (i, j) ∈ VU (13)
xtij = 0 t ∈ Xi,j, (i, j) ∈ VU (14)
y00 = 1 (15)
xtij ∈ {0, 1} (i, j) ∈ VU , t ∈ T (16)
yti ∈ {0, 1} i ∈M, t ∈ T, (17)
where γ and βi (i ∈ M ) are given coefficients. The objective function is composed by two terms: the
first term represents the total flight time between the mission way points; the second term describes the
holding duration over each mission way point as the difference between the departure and the arrival time
steps. The coefficients γ and βi make it possible to weigh the different phases of flight and holding: if
rerouting is the preferred option with respect to holding, the coefficients βi can be increased appropriately.
If γ = βi =1 (i ∈M ) the objective function minimizes the total mission duration in terms of time steps and
thus the impact of the ATC over the mission. To obtain the mission duration in time units it is necessary
to consider the duration of a generic time step ∆t. Constraints (6) and (7) are the assignment constraints,
imposing that each mission way point i has one entering arc and one leaving arc, respectively. Constraints
(8) ensure that each mission way point i has one arrival time. Constraints (9) and (10) joined with the
initial condition (15), act as Subtour Elimination Constraints and ensure the temporal coherence of the
tour. In addition they force the departure from the airport at time step t = 0. Constraints (9) ensure that if
the UAS arrives at the mission way point i at time step t, it must have departed from the previous mission
way point at an appropriate time step compatible with the flight time. Constraints (10) impose that if the
UAS leaves the mission way point i at time step t, the arrival at i must occur in a previous or in the same
14
time step. Both render the time coherence on the mission way points. These groups of constraints make
it possible to build the Hamiltonian circuit over the mission way points as in the ATSP. Constraints (14)
impose no departure, according to the minimum separation constraint, in the forbidden time steps while
constraints (13) forbid departures after the last time step. Constraints (11) and (12) impose a lower bound
(κmin) and an upper bound (κmax) on the duration of the holding over a mission way point i, ensuring that
the holding is consistent with the UAS performances. As an alternative, it is possible to set κmin equal to 0,
solve the problem and replace the corresponding holdings by using different avoidance manoeuvres such
as speed control. For example, a holding smaller than a lower bound value can be avoided by reducing the
UAS speed in the related sub-path.
Even though they are not necessary for the elimination of the sub-tours, the following inequalities can
be added to formulation (5)-(17) in order to strengthen its Linear Programming (LP) relaxation:∑t∈T
∑i∈Sub,j∈M\Sub
xtij ≥ 1 2 ≤ |Sub| ≤ |M | − 2, Sub ⊂M (18)
Such inequalities are a time extension of the well known Subtour Elimination Constraints (SEC) of the
Dantzig-Fulkerson-Johnson model ([5]) for the Asymmetric TSP (ATSP). The Padberg-Rinaldi ([14])
separation procedure can be efficiently used to separate in polynomial time such inequalities. Thus a
Branch-and-Cut algorithm was developed based on formulation (5)-(18). Before starting the algorithm, a
preprocessing phase is performed. This phase consists of two steps: computation of an upper bound on
the mission duration and a Variable Fixing procedure.
5.2. Variable Fixing procedure
The Variable Fixing procedure is based on the concept of ”infeasible departure” time steps. An ”In-
feasible Departure ” time step is a time step at which a departure of the UAS cannot take place; it is due to
three reasons. The first one depends on the first time step at which it is possible to reach a given mission
way point i. If tspi is the first time step at which it is possible to arrive at i, no departure from i to any other
mission way point can take place before tspi . The shortest path time from the airport (mission way point
0) to a mission way point i makes it possible to define its related tspi . The Dijkstra algorithm ([6]) is an
effective algorithm that is able to find the shortest path between the airport and each other mission way
point. Note that in this context a shortest path algorithm must take into consideration the presence of air
traffic. The cost of an arc in the Dijkstra algorithm has to consider two components: the holding until the
first available time step and the flight time. The Dijkstra algorithm has been properly adapted to satisfy
15
this constraint and the shortest path from the airport to each mission way point has been found in a very
short computing time. Moreover, the presence of air traffic is the second reason that makes a departure
infeasible. The associated variables correspond to those considered in constraints (14) of the formulation
(5)-(17). Finally no arrival at the airport can take place before the lower bound provided by the value of
the optimal solution of the ATSP associated with the graph GU . All these variables assume a value equal
to zero in any solution of the problem. For this reason they are removed from the model in order to reduce
its dimension significantly.
5.3. The Branch and Cut Algorithm
In this section we describe the main features of the proposed Branch and Cut algorithm (BC). Once
the Variable Fixing procedure has been executed, the algorithm starts by solving the LP relaxation of
model (5)-(17) and generating valid inequalities (18). If its solution is integer, an optimal solution has
been found; otherwise an enumeration tree is constructed. At each node of such tree the algorithm tries to
generate violated valid inequalities, by using the Padberg-Rinaldi Separation Procedure; once found, the
inequalities are added to the current LP relaxation. The inequalities added at any node of the tree are valid
for all the other nodes because inequalities (18) are valid for the formulation (5)-(17). Thus, at a given
node, all the inequalities so far generated are incorporated in the lower bound computation. This Branch-
and-Cut algorithm has been implemented using Ilog Cplex 12.1 with the default branching scheme. Even
though this algorithm is able to solve to optimality many instances of the TDTSPPCA (see Section (7)),
the time dependence of the decision variables leads to weak lower bounds.
6. Penalty-Based Model
To avoid the weakness of the time dependent formulation, the TDTSPPCA is reformulated as a TSP
with Penalties associated with the routes. These penalties represent an alternative way to model the hold-
ings necessary to avoid the conflicts.
6.1. Mathematical Formulation and main features
For a given mission way point i ∈M , let Hi be the set of all possible sequences of mission way points
and holdings that present i as last reachable mission way point, i.e. the partial route that, starting from the
airport, define a departure from i in a conflicting time step. Such conflict could be optimally avoided with
a sequence of previous holdings: since the air traffic is known a priori, once a partial route is fixed, it is
possible to find its optimal holdings distribution. Using this property, the new mathematical formulation
16
extends the Dantzig-Fulkerson-Johnson TSP model ([5]) considering all possible conflicting partial routes
and their related ”penalty” constraints. Finally, denoting with Hi the sequence of mission way points in
Hi, we can define the coefficient kh, i.e. the optimal holding of the element h of a conflicting partial route.
Note that, in this context, a sequence of mission way points conflicting in i is a part of a Hamiltonian
circuit and that it includes all mission way points until the successor of i since the holding duration is
related to an arc and not only to a mission way point.
Two types of decision variables are used. Binary variable xij ((i, j) ∈ VU ) takes value 1 if the UAS
uses the sub route (i, j) and 0 otherwise. Continuous variable pi represents the holding over the mission
way point i (i ∈M ). The MILP model of the TSPP reads as follows:
min γ∑
(i,j)∈VU
fijxij +∑i∈M
βipi (19)
∑j∈M
xji = 1 i ∈M (20)∑j∈M
xij = 1 i ∈M (21)∑i∈Sub,j∈M\Sub
xij ≥ 1 2 ≤ |Sub| ≤ |M | − 2, Sub ⊆M (22)
∑(lj)∈Hi
(kh − kmin)(1− xlj) + ph ≥ kh h ∈ Hi, Hi ∈ Hi, i ∈M (23)
xij ∈ {0, 1} (i, j) ∈ VU (24)
pi ∈ [kmin, kmax] i ∈M (25)
As in (5), the first term of the objective function represents the total flight time through the mission
way points; the second term describes the global weighted holding over all the visited mission way points
as a ”penalty” that has to be added to the total cost of the route. To this end, it is necessary to add the
”penalty” constraints (23) to the standard constraints (20) -(22)) of the Dantzing-Fulkerson-Johnson ATSP
formulation.(see [5] for further details).
Constraints (23) relate the route and the penalties (holdings). They enumerate all possible routes that could
lead the UAS to a conflict. Clearly such constraints are exponential in number, but they can be separated
using a specific procedure (see Section 6.2). To compute the coefficient kh for each h of a given sequence
Hi, it is necessary to solve the associated Holding Assignment Problem (HAP). In the specific case of not
weighted holdings, it is sufficient to wait only at the last mission way point before the conflict until the
first available departure time step.
17
6.2. Holding Assignment Problem
Given a conflicting sequence Hi and the amount of holding time steps over its last reachable mission
way point i, the goal of the HAP is to find the best way to assign such holding time steps over the mission
way points until i. We define K as a set of the holding time steps that have to be assigned. By reassigning
the holdings, the departure and arrival time steps of the following mission way points change. Thus, these
time steps could correspond to conflicting time steps. As a consequence, it is necessary to consider the
presence of further conflicts that can occur by changing the holdings. Nevertheless this solution could be
better than that associated with smaller holdings over more sensitive mission way points. To model this
kind of conflicts, further penalties are added in a manner similar to that used for the Penalty-Based model.
For a given mission way point i ∈ M and the corresponding conflicting sequence Hi, we define Q as the
set of all possible assignments of |K| holding time steps over the mission way points until i. A generic
assignment q ∈ Q can be represented by the appropriate pairs (k, h), with k ⊆ K and h ∈ Hi. Let kqh
be the minimum holding duration over h for the assignment q, i.e. the time steps interval until the first
available departure time step given the partial route Hi. We introduce a binary variable zk,h with value 1 if
k holding time steps are assigned to the mission way point h and value 0 otherwise. Moreover we use the
continuous variable p′h to represent the further holding over h, and the binary variable uh that takes value
1 if the holding h is used and value 0 otherwise. Given i ∈ M , Hi, K and Q, the corresponding MILP
model reads as follow:
min∑k∈K
∑h∈Hi
βhkzk,h +∑h∈Hi
βhp′h (26)
18
∑k∈K
∑h∈Hi
kzk,h +∑h∈Hi
p′h ≥ |K| (27)
uh ≥ zk,h k ∈ K,h ∈ Hi (28)∑k∈K
kzk,h + p′h ≤ kmax h ∈ Hi (29)∑k∈K
kzk,h + p′h ≥ kminuh h ∈ Hi (30)∑(k,h)∈q
B1k(1− zk,h) + p′h ≥ kqh h ∈ Hi, q ∈ Q (31)
zk,h ∈ {0, 1} k ∈ K,h ∈ Hi (32)
uh ∈ {0, 1} h ∈ Hi (33)
p′h ∈ [0, kmax] h ∈ Hi (34)
For the considered mission way point i ∈ M , the objective function (26) minimizes the total cost
of the associated holdings by considering two components: the holdings due to the reassignment and the
holdings corresponding to the presence of further conflicts. Constraints (27) ensures that the global holding
duration is consistent with the number of time steps that have to be assigned; Constraints (28) ensure that
the appropriate variables uh are activated. Constraints (29) and (30) represent the capacity constraints
related, respectively, to the upper bound and the lower bound on the holding duration. Constraints (31)
define the penalties by relating them with the reassignment; B1 is a large number that can be set to kqh
divided by k times the number of pairs (k, h) in q. To solve the HAP we separate the constraints (31) that
are exponential in number. The algorithm starts by solving the model (26)-(34) without any constraint
(31): if the solution does not imply further conflicts, the best solution is found. Otherwise, in each mission
way point where a further conflict occurs the related holding until the first available time step is evaluated.
Then the appropriate penalty kqh is added to the problem through the constraints (31) and the problem
is re-solved. The algorithm iterates until a non-conflicting solution is found. The optimal values of the
variables zk,h (k ∈ K,h ∈ Hi) allow then to compute the set of coefficients kh for each h ∈ Hi as follows:
kh =∑k∈K
k zk,h
6.3. Cutting Plane Algorithm
In this section we describe the main features of the Cutting Plane algorithm (CP) proposed. The
algorithm starts solving the model (19)-(25) without any constraints (23). This model corresponds to
the ”TSP − relaxation” of the problem. If the route found presents no conflict the optimal route for
the TDTSPPCA is the TSP route. Otherwise the route can be followed until the last reachable mission
19
way point without conflicts, identifying in this manner a partial sequence of mission way points. Then
the related HAP is solved using the procedure described in Section 6.2 in order to find the best holding
distribution for the considered subsequence. The resulting holdings are imposed as penalties adding the
appropriate constraints (23) and the problem is re-solved. This procedure is iteratively repeated until a
non-conflicting route is found.
7. Computational Results
We tested the algorithms proposed in the previous sections on a real air traffic scenario: the TMA
(Terminal Manoeuvring Area) of Milano Linate (ICAO code LIML), an important airport in the North of
Italy with an average of 450 air traffic movements per day. Computer Graphic tools have been used to
model the air traffic scenario, to analyze the results and to rapidly evaluate the performances of the pro-
posed algorithms. In this environment, Navigation Points such as Radio-Assistance and Fix Points (radial
and distance by radio assistance) are reported; SID (Standard Instrumental Departure) routes and STAR
(STandard Arrival Route) of the airport are modeled using graphic tools.
The air traffic data represent the position and the altitude of departing, arriving and overflying aircraft
during five different time periods. The days considered were: the 11th of November 2009 from 05:30
until 13:30 UTC, the 22nd of August 2010 from 04:30 to 12:30 UTC, the 25th of August 2010 from 04:30
to 12:30 UTC, the 25th of August 2010 from 12:30 to 20:30 UTC and the 26th of August from 04:30 to
12:30 UTC.
We considered two types of instances: Medium-Short and Medium-Long. The first type consists of visit-
ing up to 20 mission way points, whose coordinates are randomly selected among the Navigation Points
located within a circle having center in LIML and ray of 20NM (Nautical Miles). The second type consists
of visiting up to 40 mission way points whose coordinates are randomly selected among the Navigation
Points located within a circle having center in LIML and ray of 80NM. The costs of the holdings are
assigned using the following considerations: if 1 is the unit time cost of the flight phase (γ = 1), a ground
holding (in the airport) unit time cost is equal to 2. Instead, the airborne holdings have a cost decreasing
according to their distance from the airport. The basic idea is that a UAS holding near the airport requires
a more difficult management than a UAS holding far from the airport, because the former zone is more
sensitive since the traffic is more concentrated.
To model a realistic update of the air traffic, a time step of one minute has to be set: this leads to a large
number of time steps (more than 200) for a Medium-Long instances. The position and the altitude of
each aircraft are updated at each time step. As reference, we use a UAS with performances similar to the
20
General Atomic MK9 Reaper, called Predator B. We set the cruise speed to 180 Knots and the maximum
climb/descent rate to 1500 ft/min. Moreover we set kmin to 0 and kmax to 20 minutes.
As an example, the Medium-Long instance of the 11th of November 2009 is reported in Figures 3(a) and
3(b). In Figure 3(a) the TSP route is shown. The UAS is represented by a black triangle while the aircraft
are represented by a white triangle. Dashed segments represent the aircraft routes and continuous seg-
ments represent the UAS route. In this case the time required for the UAS to overfly all 20 mission way
points is 162 minutes (TSP optimal solution value) and 4 conflicts between the UAS and the aircraft are
detected (represented by black bullets, one of them is covered by the flight SMX5294). In Figure 3(b) the
route without conflicts is reported. The duration of the mission in the controlled air space is 171 minutes:
a rerouting makes it possible to avoid the piloted air traffic.
An interesting example of UAS conflict resolution in such mission is reported in Figure 4. The points
AMOXI, LIMBA and DIXER are lined up to Runway 36 of Linate; the arriving aircraft coming from
South follow this route. Looking at the figures in sequence it is possible to recognize an arrival sequence
concerning flights AZA2036, ACL324 and AZA2032. The UAS route includes the path between DIXER
and LIMBA in the opposite direction. This path is performed by the UAS between ACL324 and AZA2032
without separation minima infringement.
Tables 1 and 2 report the main characteristics and the results of the heuristic algorithms in the Medium-
Short and Medium-Long instances, respectively. The name of an instance is composed by: the ICAO code
of the Airport (LIML), the date, the simulation beginning time plus ”S” or ”L” for Medium-Short or
Medium-Long instances respectively and the number of mission way points to visit (m). The other char-
acteristics are: the number of simulated aircraft (n), the value in minutes of the TSP solution (Lower
Bound on the value of the optimal solution in Controlled Air Space) and the number of conflicts on the
TSP route. Then the tables report the heuristic solutions obtained by HTSP and by NN: for each of them
two objective values and the related percentage gaps are reported. The first one (val) is the value of the
solution found by the heuristic and the second one (LS) is the value of the Local Search applied to such
solution. Finally the percentage gaps are computed between these upper bounds and the corresponding
optimal solution value; for each group of instances the average gaps are also reported.
Table 3 reports the comparison between the Branch-and-Cut (BC) and the Cutting-Plane (CP) algorithms
on the Medium-Short instances. The first two columns report the instance name and the optimal solution
value found by both algorithms. Then we report the results corresponding to the BC algorithm. In par-
ticular, we report two lower bounds (LB1 and LB2) with the corresponding computing time. LB1 is the
21
(a) TSP route with conflicts
(b) Optimal route without conflicts
Figure 3: Example of routes
22
(a) (b)
(c) (d)
Figure 4: Example of Avoidance
23
continuous relaxation value of model (5)-(17) while LB2 is the continuous relaxation value of the same
model with variable reduction and valid inequalities (18). Column T reports the computing time of the
BC algorithm with a time limit of 1200 seconds (only in one case the algorithm was not able to solve
the instance to proven optimality); then two columns report the number of nodes (Nodes) and the number
of cuts (Cuts). Finally the table reports the solution time (T) and the number of iterations (It) of the CP
algorithm. For each group of instances the average values are reported. Table 3 shows that on average it
is better to use the CP algorithm for this group of instances but in two cases the computing time of the
BC algorithm is lower. Table 4 reports the results on the Medium-Long Mission instances for only the
CP algorithm, since the BC algorithm is not effective in these cases. In columns 2, 3 and 4 we report,
respectively, the optimal solution value, the relative computing time and the number of iterations needed.
In order to obtain good feasible solutions within short computing time, we executed the algorithm CP with
a time limit equal to 1200 seconds. Since if the time limit is reached the corresponding solution is not
feasible, we obtain a feasible solution by following the solution found until the first conflict is detected,
and then we apply the heuristic algorithms HTSP and NN and the local search procedure (LS). Columns
5 and 6 report the lower bound found by the algorithm obtained within the time limit of 1200 seconds
and the corresponding gap for the instances not solved to optimality within that time limit. For these 11
instances, the last four columns of the table (CP 1200sec + Heur) report the values and the gaps achieved
applying the heuristic algorithms (without and with the local search procedure LS) to the solution found
within the time limit (as described in Tables 1 and 2). In other words, in case of time limit, the infeasible
solution found by the algorithm CP is completed using both HTSP and NN algorithms. The best value
obtained among them is reported in column ”val” and the corresponding solution is used as input for the
Local Search procedure (LS).
8. Future developments
Future works will focus on the extension to the dynamic case, on the use of other avoidance techniques
and on the case of multi-UAS missions. In the dynamic case, the air traffic situation is not known in
advance; however in a given step is known, for each aircraft, the route cleared by the air traffic controllers.
In this way, a reliable and realistic forecast of the air traffic for the next few minutes is available and can
be used for the planning of the next mission way point to be visited by the UAS. Moreover, the algorithms
used for the static case, considered in the planning phase, can be easily extended to the dynamic case
by defining two subsets of mission way points at each time step: the subset of points already visited and
the subset of points that have to be visited. At a given time step, once the decision variables of the first
24
HTSP NN
Instance n TSP conf val gap LS gap val gap LS gap
LIML 11/11 05:30 S - 10 32 29 2 43 38.71 43 38.71 31 0.00 31.00 0.00
LIML 22/08 04:30 S - 10 28 29 0 29 0.00 29 0.00 51 75.86 47.00 62.07
LIML 25/08 04:30 S - 10 40 29 5 42 44.83 38 31.03 35 20.69 35.00 20.69
LIML 25/08 12:30 S - 10 34 29 1 34 9.68 34 9.68 31 0.00 31.00 0.00
LIML 26/08 04:30 S - 10 38 29 2 58 81.25 58 81.25 63 96.88 54.00 68.75
34.9 32.1 38.7 30.3
LIML 11/11 05:30 S - 15 68 31 3 38 18.75 36 12.50 37 15.63 37.00 15.63
LIML 22/08 04:30 S - 15 61 31 1 37 8.82 37 8.82 37 8.82 37.00 8.82
LIML 25/08 04:30 S - 15 74 31 1 34 0.00 34 0.00 40 17.65 40.00 17.65
LIML 25/08 12:30 S - 15 71 31 2 38 15.15 33 0.00 36 9.09 35.00 6.06
LIML 26/08 04:30 S - 15 72 31 1 37 8.82 37 8.82 37 8.82 37.00 8.82
10.3 6.0 12.0 11.4
LIML 11/11 05:30 S - 20 123 41 1 52 26.83 41 0.00 42 2.44 42.00 2.44
LIML 22/08 04:30 S - 20 117 41 3 49 13.95 46 6.98 43 0.00 43.00 0.00
LIML 25/08 04:30 S - 20 131 41 3 63 50.00 50 19.05 48 14.29 48.00 14.29
LIML 25/08 12:30 S - 20 127 41 0 41 0.00 41 0.00 42 2.44 42.00 2.44
LIML 26/08 04:30 S - 20 125 41 3 67 59.52 42 0.00 43 2.38 43.00 2.38
30.1 5.2 4.3 4.3
Table 1: The table reports the main characteristics and the heuristic algorithm results for the Medium-Short Mission instances
subset are set to the values corresponding to the part of the route already flown, the problem is solved by
determining the next point among the points of the second subset. This procedure is repeated until all the
points have been visited. The main assumption of this approach is that the used algorithm must be able to
solve the problem within a computing time smaller than the traffic updating interval. Tactical avoidance
techniques will be modeled and used in case of changes in the aircraft routes or in the controller clearances
that affect the route until the next mission way point (this corresponds to the route in progress that can not
be changed using this methodology). The use of this approach based on the two subsets of points, will
also allow changing tactically the point to be visited, or including new points or changing the position of
the old ones. Rolling horizon techniques could be another approach to be investigated for the extension to
the dynamic case. Thanks to temporal discretization supported by our approach, the algorithm can also be
extended in order to consider moving targets (e.g., tracking animals or ships) or targets dependent upon
weather conditions. Finally the uncertainty can be captured in the arc costs, e.g., giving higher costs to
arcs which are likely to produce conflicts with the air traffic.
25
HTSP NN
Instance n TSP conf val gap LS gap val gap val gap
LIML 11/11 05:30 L - 10 32 55 2 68 6.25 67 4.69 86 34.38 86.00 34.38
LIML 22/08 04:30 L - 10 28 55 3 68 17.24 64 10.34 80 37.93 80.00 37.93
LIML 25/08 04:30 L - 10 40 55 3 63 14.55 63 14.55 90 63.64 85.00 54.55
LIML 25/08 12:30 L - 10 34 55 3 80 23.08 70 7.69 73 12.31 71.00 9.23
LIML 26/08 04:30 L - 10 38 55 3 68 19.30 67 17.54 85 49.12 85.00 49.12
16.1 11.0 39.5 37.0
LIML 11/11 05:30 L - 15 68 98 1 117 15.84 105 3.96 128 26.73 128.00 26.73
LIML 22/08 04:30 L - 15 61 98 2 119 16.67 111 8.82 122 19.61 120.00 17.65
LIML 25/08 04:30 L - 15 74 98 2 100 1.01 100 1.01 126 27.27 126.00 27.27
LIML 25/08 12:30 L - 15 71 98 2 117 11.43 110 4.76 127 20.95 127.00 20.95
LIML 26/08 04:30 L - 15 72 98 1 106 7.07 104 5.05 122 23.23 121.00 22.22
10.4 4.7 23.6 23.0
LIML 11/11 05:30 L - 20 123 162 4 189 10.53 189 10.53 197 15.20 197.00 15.20
LIML 22/08 04:30 L - 20 117 162 4 175 6.06 175 6.06 202 22.42 189.00 14.55
LIML 25/08 04:30 L - 20 131 162 4 192 15.66 176 6.02 204 22.89 204.00 22.89
LIML 25/08 12:30 L - 20 127 162 4 220 27.91 194 12.79 187 8.72 187.00 8.72
LIML 26/08 04:30 L - 20 125 162 3 204 24.39 181 10.37 201 22.56 190.00 15.85
16.9 9.2 18.4 15.4
LIML 11/11 05:30 L - 25 211 176 5 190 2.15 190 2.15 220 18.28 220.00 18.28
LIML 22/08 04:30 L - 25 204 176 2 204 10.27 204 10.27 221 19.46 221.00 19.46
LIML 25/08 04:30 L - 25 216 176 4 192 4.35 192 4.35 220 19.57 196.00 6.52
LIML 25/08 12:30 L - 25 212 176 6 223 20.54 223 20.54 232 25.41 210.00 13.51
LIML 26/08 04:30 L - 25 214 176 3 204 11.48 204 11.48 211 15.30 211.00 15.30
9.8 9.8 19.6 14.6
LIML 11/11 05:30 L - 30 213 179 5 214 12.63 214 12.63 229 20.53 229.00 20.53
LIML 22/08 04:30 L - 30 207 179 4 207 9.52 207 9.52 233 23.28 233.00 23.28
LIML 25/08 04:30 L - 30 220 179 3 203 9.73 203 9.73 215 16.22 211.00 14.05
LIML 25/08 12:30 L - 30 214 179 5 221 13.92 221 13.92 216 11.34 212.00 9.28
LIML 26/08 04:30 L - 30 216 179 4 205 8.47 205 8.47 211 11.64 211.00 11.64
10.9 10.9 16.6 15.8
LIML 11/11 05:30 L - 35 229 191 6 225 14.80 218 11.22 232 18.37 231.00 17.86
LIML 22/08 04:30 L - 35 226 191 4 236 21.65 236 21.65 253 30.41 253.00 30.41
LIML 25/08 04:30 L - 35 271 191 4 216 9.09 216 9.09 244 23.23 231.00 16.67
LIML 25/08 12:30 L - 35 245 191 7 253 30.41 220 13.40 253 30.41 253.00 30.41
LIML 26/08 04:30 L - 35 250 191 2 228 14.57 225 13.07 244 22.61 244.00 22.61
18.1 13.7 25.0 23.6
LIML 11/11 05:30 L - 40 247 199 9 228 9.09 228 9.09 259 23.92 258.00 23.44
LIML 22/08 04:30 L - 40 232 199 6 262 27.18 251 21.84 280 35.92 280.00 35.92
LIML 25/08 04:30 L - 40 282 199 6 220 6.80 220 6.80 263 27.67 263.00 27.67
LIML 25/08 12:30 L - 40 257 199 10 226 10.78 224 9.80 263 28.92 253.00 24.02
LIML 26/08 04:30 L - 40 261 199 4 232 12.62 230 11.65 250 21.36 247.00 19.90
13.3 11.8 27.6 26.2
Table 2: The table reports the main characteristics and the heuristic algorithm results for the Medium-Long Mission instances
26
BC CP
Instance opt LB1 T1 LB2 T2 T Nodes Cuts T It.
LIML 11/11 05:30 S - 10 31 26 0.0 29 0.0 3.8 38 6 6.7 34
LIML 22/08 04:30 S - 10 29 26 0.1 29 0.0 1.5 0 6 0.1 1
LIML 25/08 04:30 S - 10 29 26 0.0 29 0.0 0.9 0 7 0.9 8
LIML 25/08 12:30 S - 10 31 26 0.0 29 0.0 5.2 30 6 19.5 53
LIML 26/08 04:30 S - 10 32 26 0.1 29 0.0 104.8 1290 37 71.2 136
23.2 271.6 12.4 19.7 46.4
LIML 11/11 05:30 S - 15 32 28 0.2 32 0.0 49.3 0 6 0.2 2
LIML 22/08 04:30 S - 15 34 28 0.3 32 0.0 516.4 273 23 33.4 58
LIML 25/08 04:30 S - 15 34 28 0.2 32 0.0 1200.0 1233 75 16.7 47
LIML 25/08 12:30 S - 15 33 28 0.1 32 0.0 103.1 41 6 2.7 12
LIML 26/08 04:30 S - 15 34 28 0.2 32 0.0 356.0 266 33 31.4 58
445.0 362.6 28.6 16.9 35.4
LIML 11/11 05:30 S - 20 41 37 0.4 41 0.1 65.5 578 32 0.6 2
LIML 22/08 04:30 S - 20 43 37 0.4 42 0.1 962.5 2864 44 69.7 118
LIML 25/08 04:30 S - 20 42 36 0.3 42 0.1 32.7 259 23 6.3 15
LIML 25/08 12:30 S - 20 41 37 0.3 41 0.1 368.3 5089 53 0.1 1
LIML 26/08 04:30 S - 20 42 37 0.3 42 0.1 89.3 316 29 1.2 4
303.6 1821.2 36.2 15.6 28.0
Table 3: The table reports the comparison between BC algorithm and CP algorithm for the Medium-Short Mission instances.
9. Conclusions
In this paper we study the Time Dependent Traveling Salesman Planning Problem in Controlled
Airspace. We have proposed and compared two different MILP models and designed two exacts algo-
rithms to solve them. The first one is a time dependent formulation of the problem able to model a real
update of the air traffic. Then, in order to avoid the weakness of the time dependent formulation, we
propose a second formulation based on penalties. The methodology we propose is innovative since the
conflict resolution uses an ATM approach, i.e. we model the avoidances through manoeuvres planned by
Air Traffic Controllers in order to optimize a conflict free route. Some heuristic algorithms have also been
proposed and tested. Computational results show how the proposed algorithms are able to solve real-world
instances in short computing time, which makes the proposed algorithms suitable for extensions to real
time re-optimization.
27
CP CP 1200sec CP 1200sec + Heur
Instance opt T It. lb2 gap val gap LS gap
LIML 11/11 05:30 L - 10 64 59.2 118 * * * * * *
LIML 22/08 04:30 L - 10 58 1.3 14 * * * * * *
LIML 25/08 04:30 L - 10 55 0.1 2 * * * * * *
LIML 25/08 12:30 L - 10 65 98.3 203 * * * * * *
LIML 26/08 04:30 L - 10 57 0.8 18 * * * * * *
32.0 71.0
LIML 11/11 05:30 L - 15 101 1.9 19 * * * * * *
LIML 22/08 04:30 L - 15 102 1.8 17 * * * * * *
LIML 25/08 04:30 L - 15 99 0.1 2 * * * * * *
LIML 25/08 12:30 L - 15 105 81.1 145 * * * * * *
LIML 26/08 04:30 L - 15 99 0.4 5 * * * * * *
17.1 37.6
LIML 11/11 05:30 L - 20 171 4486.0 346 170 0.58 171 0.00 171 0.00
LIML 22/08 04:30 L - 20 165 4.1 21 * * * * * *
LIML 25/08 04:30 L - 20 166 11.8 35 * * * * * *
LIML 25/08 12:30 L - 20 172 3545.0 352 171 0.58 172 0.00 172 0.00
LIML 26/08 04:30 L - 20 164 1.2 12 * * * * * *
1609.6 153.2
LIML 11/11 05:30 L - 25 186 2456.9 456 186 0.00 186 0.00 186 0.00
LIML 22/08 04:30 L - 25 185 1111.9 235 * * * * * *
LIML 25/08 04:30 L - 25 184 130.8 64 * * * * * *
LIML 25/08 12:30 L - 25 185 894.6 543 * * * * * *
LIML 26/08 04:30 L - 25 183 122.9 121 * * * * * *
943.4 283.8
LIML 11/11 05:30 L - 30 190 6518.1 1231 187 1.58 214 12.63 213 12.11
LIML 22/08 04:30 L - 30 189 1176.2 632 * * * * * *
LIML 25/08 04:30 L - 30 185 45.1 81 * * * * * *
LIML 25/08 12:30 L - 30 194 2843.4 1987 188 3.09 221 13.92 221 13.92
LIML 26/08 04:30 L - 30 189 1124.2 427 * * * * * *
2341.4 871.6
LIML 11/11 05:30 L - 35 196 326.4 181 * * * * * *
LIML 22/08 04:30 L - 35 194 29.2 48 * * * * * *
LIML 25/08 04:30 L - 35 198 11546.3 771 194 2.02 218 10.10 215 8.59
LIML 25/08 12:30 L - 35 194 11.1 22 * * * * * *
LIML 26/08 04:30 L - 35 199 4455.1 507 193 3.02 220 10.55 204 2.51
3273.6 305.8
LIML 11/11 05:30 L - 40 209 123327.4 2931 204 2.39 228 9.09 228 9.09
LIML 22/08 04:30 L - 40 206 18408.3 949 203 1.46 262 27.18 242 17.48
LIML 25/08 04:30 L - 40 206 17140.3 937 203 1.46 220 6.80 206 0.00
LIML 25/08 12:30 L - 40 204 114.0 97 * * * * * *
LIML 26/08 04:30 L - 40 206 49235.2 867 201 2.43 232 12.62 232 12.62
41645.0 1156.2
Table 4: Table reports the results of the CP algorithm for the Medium-Long Mission instances.
28
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